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<span class="kw">set </span><font color="Maroon" title="c1">K</font> =  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="hermitan.html#NM2" title="HERMITAN:NM.2">sesquilinear-Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> holds <br/> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="hermitan.html#NM2" title="HERMITAN:NM.2">sesquilinear-Form</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">wq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">wqr</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">vql</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">rlf</font> =  <a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">qf</font> =  <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">lrf</font> =  <a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:122"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:122_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
for <font color="Olive" title="b1">ww</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Olive" title="b1">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c14">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c14">ww</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T64" target="_self" title="HERMITAN:th.64">Th64</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c14">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c15">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:122_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:122"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_2"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:122_2"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
for <font color="Olive" title="b1">ww</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Olive" title="b1">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c14">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c14">ww</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T64" target="_self" title="HERMITAN:th.64">Th64</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c14">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c15">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_2"><span class="lab"><font color="Green" title="E60">A4</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:122_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_2"><span class="lab"><font color="Green" title="E59">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:122"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_3"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E4:122_3"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
for <font color="Olive" title="b1">vv</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">w</font> = <font color="Maroon" title="c13">ww</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T64" target="_self" title="HERMITAN:th.64">Th64</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c14">w</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_3"><span class="lab"><font color="Green" title="E60">A6</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:122_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_3"><span class="lab"><font color="Green" title="E59">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:122"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_4"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E4:122_4"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
for <font color="Olive" title="b1">vv</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">w</font> = <font color="Maroon" title="c13">ww</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T64" target="_self" title="HERMITAN:th.64">Th64</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c14">w</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">vv</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_4"><span class="lab"><font color="Green" title="E60">A8</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:122_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_4"><span class="lab"><font color="Green" title="E59">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:122"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_5"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:122_5"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
for <font color="Olive" title="b1">ww</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Olive" title="b1">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">v</font> = <font color="Maroon" title="c13">vv</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T65" target="_self" title="HERMITAN:th.65">Th65</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c15">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_5"><span class="lab"><font color="Green" title="E60">A10</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:122_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_5"><span class="lab"><font color="Green" title="E59">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:122"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_6"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">vv</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_6"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">vv</font>
 <span class="kw">and </span><br/><a NAME="E4:122_6"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
for <font color="Olive" title="b1">ww</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Olive" title="b1">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">v</font> = <font color="Maroon" title="c13">vv</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T65" target="_self" title="HERMITAN:th.65">Th65</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">v</font>,<font color="Maroon" title="c15">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c13">vv</font>,<font color="Maroon" title="c15">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_6"><span class="lab"><font color="Green" title="E60">A12</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:122_6"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_6"><span class="lab"><font color="Green" title="E59">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:122"/>
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:122_7"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:122_7"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E4:122_7"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
for <font color="Olive" title="b1">vv</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">vv</font>,<font color="Maroon" title="c13">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">vv</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T65" target="_self" title="HERMITAN:th.65">Th65</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">vv</font>,<font color="Maroon" title="c13">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:122_7"><span class="lab"><font color="Green" title="E60">A14</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:122_7"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:122_7"><span class="lab"><font color="Green" title="E59">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E8:122"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">ww</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:122"/><span class="lab"><font color="Green" title="E59">A15</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ww</font>
 <span class="kw">and </span><br/><a NAME="E11:122"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
for <font color="Olive" title="b1">vv</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">vv</font>,<font color="Maroon" title="c13">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">vv</font>,<font color="Maroon" title="c13">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">vv</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T65" target="_self" title="HERMITAN:th.65">Th65</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c14">vv</font>,<font color="Maroon" title="c13">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17" title="BILINEAR:func.17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" title="HERMITAN:func.7">RQ*Form</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c13">ww</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T66" target="_self" title="HERMITAN:th.66">Th66</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122"><span class="lab"><font color="Green" title="E60">A16</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:122"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:122"><span class="lab"><font color="Green" title="E59">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
